Smoothing problem (stochastic processes): Difference between revisions

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{{technical|date=November 2017}}

The '''smoothing problem''' (not to be confused with [[smoothing]] in [[statistics]], [[image processing]] and other contexts) is the problem of [[density estimation|estimating]] an unknown [[probability density function]] recursively over time using incremental incoming measurements. It is one of the main problems defined by [[Norbert Wiener]].<ref name="wiener-report">1942, ''Extrapolation, Interpolation and Smoothing of Stationary Time Series''. A war-time classified report nicknamed "the yellow peril" because of the color of the cover and the difficulty of the subject. Published postwar 1949 [[MIT Press]]. http://www.isss.org/lumwiener.htm {{Webarchive|url=https://web.archive.org/web/20150816041622/http://www.isss.org/lumwiener.htm |date=2015-08-16 }}</ref><ref name="wiener-book">{{cite book |last=Wiener, |first=Norbert (|author-link=Norbert Wiener |year=1949). |title=Extrapolation, Interpolation, and Smoothing of Stationary Time Series.: NewWith York:Engineering Wiley.Applications {{ISBN|0url=https://direct.mit.edu/books/oa-262monograph/4361/Extrapolation-73005Interpolation-7and-Smoothing-of |publisher=[[MIT Press]] |isbn=9780262257190}}.</ref> A '''smoother''' is an algorithm that implements a solution to this problem, typically based on [[recursive Bayesian estimation]]. The smoothing problem is closely related to the [[filtering problem]], both of which are studied in Bayesian smoothing theory.

A smoother is often a two-pass process, composed of forward and backward passes. Consider doing estimation (prediction/retrodiction) about an ongoing process (e.g. tracking a missile) based on incoming observations. When new observations arrive, estimations about past needs to be updated to have a smoother (more accurate) estimation of the whole estimated path until now (taking into account the newer observations). Without a backward pass (for [[retrodiction]]), the sequence of predictions in an online filtering algorithm does not look smooth. In other words, retrospectively, it is as if we are using future observations for improving estimation of a point in past, when those observations about future points become available. Note that time of estimation (which determines which observations are available) can be different to the time of the point that the prediction is about (that is subject to prediction/retrodiction). The observations about later times can be used to update and improved the estimations about earlier times. Doing so leads to smoother-looking estimations (retrodiction) about the whole path.

==Examples of smoothers ==

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== The confusion in terms and the relation between Filtering and Smoothing problems==

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ThereThe terms Smoothing and Filtering are used for four termsconcepts that causemay initially be confusionconfusing: Smoothing (in two senses: estimation and convolution), and Filtering (again in two senses: estimation and convolution).

Smoothing (estimation) and smoothing (convolution) despite being labelled with the same name in English language, can mean totally different, butmathematical soundprocedures. likeThe theyrequirements areof apparentlyproblems similar.they The conceptssolve are different. andThese concepts are useddistinguished inby almostthe differentcontext historical(signal contexts.processing Theversus '''requirements'''estimation areof verystochastic differentprocesses).

NoteThe historical reason for this confusion is that initially, the Wiener's suggested a "smoothing" filter that was just a convolution,. butLater theon his proposed solutions for obtaining a smoother estimation laterseparate developments wereas different:two onedistinct concepts. One was about attaining a smoother estimation by taking into account past observations, and the other one was smoothing using filter design in the sense of (design of a convolution filter. This is a source of confusion).

Both the smoothing problem (in sense of estimation) and the filtering problem (in sense of estimation) are often confused with smoothing and filtering in other contexts (especially non-stochastic signal processing, often a name of various types of convolution). These names are used in the context of World War 2 with problems framed by people like [[Norbert Wiener]].<ref name="wiener-report"/><ref name="wiener-book" /> One source of confusion is the [[Wiener Filter]] is in form of a simple convolution. However, in Wiener's filter, two time-series are given. When the filter is defined, a straightforward convolution is the answer. However, in later developments such as Kalman filtering, the nature of filtering is different to convolution and it deserves a different name.

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The distinction is described in the following two senses:

1. Convolution: The smoothing in the sense of '''convolution''' is simpler. For example, moving average, low-pass filtering, convolution with a kernel, or blurring using Laplace filters in [[image processing]]. It is often a [[filter design]] problem. Especially non-stochastic and non-Bayesian signal processing, without any hidden variables.

2. Estimation: The '''smoothing problem''' (or Smoothing in the sense of '''estimation''') uses Bayesian and state-space models to estimate the hidden state variables. This is used in the context of World War 2 defined by people like Norbert Wiener, in (stochastic) control theory, radar, signal detection, tracking, etc. The most common use is the Kalman Smoother used with Kalman Filter, which is actually developed by Rauch. The procedure is called Kalman-Rauch recursion.

It is one of the main problems solved by [[Norbert Wiener]].<ref name="wiener-report"/><ref name="wiener-book"/>

Most importantly, in the Filtering problem (sense 2) the information from observation up to the time of the current sample is used. In smoothing (also sense 2) all observation samples (from future) are used. Filtering is causal but smoothing is batch processing of the same problem, namely, estimation of a time-series process based on serial incremental observations.